\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 23, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/23\hfil Multiple solutions] {Multiple solutions for perturbed $p$-Laplacian problems on $\mathbb{R}^N$} \author[Y. Li \hfil EJDE-2015/23\hfilneg] {Ying Li} \address{Ying Li \newline Department of Mathematics and System Science, College of Science, National University of Defense Technology, Changsha 410073, China} \email{liying2014uci@gmail.com} \thanks{Submitted July 24, 2014. Published January 27, 2015.} \subjclass[2000]{35J20, 49J40, 58E30} \keywords{Multiple solutions; perturbed $p$-Laplacian; critical point theory; \hfill\break\indent boundary-value problem on $R^N$; variational methods} \begin{abstract} We establish the existence of at least three solutions for a perturbed $p$-Laplacian problem on $\mathbb{R}^N$. Our approach is based on variational methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this work, we show the existence of at least three solutions for the nonlinear perturbed problem \begin{equation}\label{e1} \begin{gathered} -\Delta_p u + |u|^{p-2}u = \lambda \alpha(x) f(u) + \mu \beta(x) h(u) \quad x \in \mathbb{R}^N,\\ u \in W^{1,p}(\mathbb{R}^N), \end{gathered} \end{equation} where $(\mathbb{R}^N, |\cdot|)$, $N > 1$, is the usual Euclidean space, $\Delta_p u : = \operatorname{div}(|\nabla u|^{p-2} \nabla u)$ with $p > N$, stands for the $p$-Laplacian operator, $f,h: \mathbb{R}\to \mathbb{R}$ are continuous functions, $\alpha, \beta \in L^1 (\mathbb{R}^N) \cap L^\infty (\mathbb{R}^N)$ are nonnegtive (not identically zero) radially symmetric maps, $\lambda$ is a positive real parameter and $\mu$ is a non-negative parameter. The main objective of this article is to investigate the existence and multiplicity solutions to the above elliptic equation defined on the whole space $\mathbb{R}^N$, by using variational methods. Many technical difficulties appear studying problems on unbounded domains (see \cite{MR1244943,MR695535,MR1811948,MR1017074}). For instance, unlike bounded domains, no compact embedding is available for $W^{1,p} (\mathbb{R}^N)$; although the embedding $W^{1,p} (\mathbb{R}^N ) \hookrightarrow L^\infty (\mathbb{R}^N)$ is continuous due to Morrey's theorem ($p > N$ ), it is far from being compact. However, the subspace of radially symmetric functions of $W^{ 1,p} (\mathbb{R}^N )$, denoted further by $W_r^{ 1,p} (\mathbb{R}^N )$, can be embedded compactly into $L^\infty (\mathbb{R}^N$ ) whenever $2 \leq N < p < +\infty$ as proved in \cite[Theorem 3.1]{MR2183381} (see Lemma \ref{lemma2.1}). In this article, employing a three critical points theorem obtained in \cite{MR2604276} which we recall in the next section (Theorem \ref{t1}), we ensure the existence of at least three weak solutions for the problem \eqref{e1}. The aim of this work is to establish precise values of $\lambda$ and $\mu$ for which the problem \eqref{e1} admits at least three weak solutions. Our result is motivated by the recent work of Candito and Molica Bisci \cite{MR3100074}. In that paper, problem \eqref{e1} has infinitely many radial solutions when $\mu = 0$ and $\lambda$ in a suitable interval. Theorem \ref{t1} has been used for establishing the existence of at least three solutions for eigenvalue problems in the papers \cite{MR2805489,MR2810717,MR2813211}. Fora review on the subject, we refer the reader to \cite{MR3190544}. \section{Preliminaries} Our main tool is the following three critical points theorem. \begin{theorem}[{\cite[Theorem 2.6]{MR2604276}}] \label{t1} Let $X$ be a reflexive real Banach space, $ \Phi:X \to \mathbb{R}$ be a coercive continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional whose G\^{a}teaux derivative admits a continuous inverse on $X^{*}$, and $\Psi:X\to \mathbb{R}$ be a continuously G\^{a}teaux differentiable functional whose G\^{a}teaux derivative is compact, such that $ \Phi(0)=\Psi(0)=0$. Assume that there exist $r>0$ and $\overline{x}\in X$, with $r< \Phi(\overline{x})$ such that \begin{itemize} \item[(a1)] $\frac{1}{r} \sup_{ \Phi(x)\leq r} \Psi(x) < \frac{\Psi(\overline{x})}{\Phi(\overline{x})}$, \item[(a2)] for each $\lambda\in \Lambda_{r}:= ]\frac{\Phi(\overline{x})}{\Psi(\overline{x})}, \frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}[$ the functional $ \Phi-\lambda \Psi$ is coercive. \end{itemize} Then, for each $\lambda\in\Lambda_{r}$ the functional $\Phi-\lambda \Psi$ has at least three distinct critical points in $X$. \end{theorem} The standard Sobolev space $ W^{1,p}(\mathbb{R}^N)$ is equipped with the norm $$ \|u\| := \Big(\int_{\mathbb{R}^N}|\nabla u(x)|^p dx + \int_{\mathbb{R}^N} |u(x)|^pdx\Big)^{1/p}. $$ Since by hypotheses $p > N$, $W^{1,p}(\mathbb{R}^N)$ is continuously embedded in $L^\infty (\mathbb{R}^N )$ and we obtain the following lemma. \begin{lemma}[{\cite[Remark 2.2]{MR2183381}}]\label{lem1} Let $u\in W^{1,p}(\mathbb{R}^N)$. Then \begin{equation}\label{e3} \| u\|_{\infty} \leq \frac{2p}{p-N}\|u\| \end{equation} for every $u \in W^{1,p}(\mathbb{R}^N)$. \end{lemma} We also note that, in the low-dimensional case, every function $u \in W^{1,p}(\mathbb{R}^N)$ admits a continuous representation (see \cite[p. 166]{MR697382}). In the sequel we will replace $u$ by this element. Let $O(N)$ stands for the orthogonal group of $\mathbb{R}^N$ and $B(0, s)$ denotes the open $N$-dimensional ball of center zero, radius $s > 0$, and standard Lebesgue measure , $\operatorname{meas}(B(0, s))$. Finally, we set \[ \|\alpha\|_{B(0,s/2)} := \int_{B(0,s/2)} \alpha(x) dx. \] We say that a function $u\in W^{1,p}(\mathbb{R}^N)$ is a weak solution of \eqref{e1} if \begin{align*} &\int_{\mathbb{R}^N} |\nabla u(x)|^{p-2} \nabla u(x) \cdot \nabla v(x) dx + \int_{\mathbb{R}^N} |u(x)|^{p-2} u(x) v(x)dx \\ &-\lambda\int_{\mathbb{R}^N}\alpha(x)f(u(x))v(x)dx -\mu\int_{\mathbb{R}^N}\beta(x) h(u(x))v(x) dx=0 \end{align*} for every $v\in W^{1,p}(\mathbb{R}^N)$. For completeness, we also recall here the principle of symmetric criticality that plays a central role in many problems from differential geometry and physics, and in partial differential equations. The action of a topological group $G$ on the Banach space $(X, \|\cdot\|_X )$ is a continuous map $\varsigma : G \times X \to X : (g, x) \to \varsigma(g, u) =: gu$, such that \[ 1u = u, \quad (gm)u = g(mu), \quad u \mapsto gu \text{ is linear}. \] The action is said to be isometric if $\|gu\|_X = \|u\|_X$, for every $g \in G$. Moreover, the space of $G$-invariant points is defined by \[ \operatorname{Fix}(G) := {u \in X : gu = u, \forall g \in G}, \] and a map $m : X \to \mathbb{R}$ is said to be $G$-invariant if $m \circ g = m$ for every $g \in G$. \begin{theorem}[Palais (1979)] Assume that the action of the topological group $G$ on the Banach space $X$ is isometric. If $J \in C^1 (X; \mathbb{R})$ is $G$-invariant and if $u$ is a critical point of $J$ restricted to Fix($G$), then $u$ is a critical point of $J$. \end{theorem} The action of the group $O(N )$ on $W^{ 1,p} (\mathbb{R}^N )$ can be defined by $(gu)(x) := u(g^{-1} x)$, for every $g \in W^{ 1,p} (\mathbb{R}^N )$ and $x \in \mathbb{R}^N$. It is clear that this group acts linearly and isometrically, which means $\|u\| = \|gu\|$, for every $g \in O(N )$ and $u \in W^{ 1,p} (\mathbb{R}^N )$. Defining the subspace of radially symmetric functions of $W^{ 1,p} (\mathbb{R}^N )$ by \[ X := W_r^{ 1,p} (\mathbb{R}^N ) := \{u \in W^{ 1,p} (\mathbb{R}^N ) : gu = u, \forall g \in O(N )\}, \] we can state the following crucial embedding result due to Krist\'{a}ly and principally based on a Strauss-type estimation (see \cite{MR0454365}). \begin{lemma}\label{lemma2.1} The embedding $W_r^{ 1,p} (\mathbb{R}^N ) \hookrightarrow L^\infty (\mathbb{R}^N )$, is compact whenever $2 \leq N < p < +\infty$. \end{lemma} See \cite[Theorem 3.1]{MR2183381} for details. We also cite a recent monograph by Krist\'{a}ly, R\u{a}dulescu and Varga \cite{MR2683404} and the classical book of Willem \cite{MR1400007} as a reference for these topics. For the sake of convenience, we define \[ F(t)=\int_{0}^{t}f(\xi)d\xi\quad \text{for all }t\in \mathbb{R},\quad H(t)=\int_{0}^{t}h(\xi)d\xi\quad \text{for all }t\in\mathbb{R}. \] \section{Main results} Fix $\tau > 0$ such that \begin{equation}\label{eq:3.1} \kappa := \frac{\|\alpha\|_{B(0,\tau/2)}}{\omega_\tau \big( \frac{2p}{p-N} \big)^p \big\{ \frac{\sigma(N,p)}{\tau^p} + l(p,N) \big\} \|\alpha\|_1} > 0 \end{equation} where $\sigma(N, p) := 2^{ p -N} (2^N - 1)$, as well as \[ l(p,N) := \frac{1 + 2^{N + p} N B_{(1/2,1)} (N,p+1)}{2^N} \] in which $B_{ (1/2,1)} (N, p + 1)$ denotes the generalized incomplete beta function defined as follows: \[ B_{(1/2,1)}(N,p+1) := \int_{1/2}^1 t^{N-1} (1-t)^p dt. \] We also note that $\omega_\tau := \operatorname{meas}(B(0, \tau )) = \tau^N \frac{\tau^{ N/2}}{\Gamma ( 1 + \frac{N}{2})}$, where $\Gamma$ is the Gamma function defined by \[ \Gamma(t) := \int_0^{+\infty} z^{t-1} e^{-z} dz \quad (\forall t > 0). \] To introduce our result, we fix three constants $c > 0$ and $\zeta$ such that $$ \frac{1}{\kappa F(\zeta)} > \frac{c^p}{ \sup_{|t|\leq c} F(t)}\,. $$ Taking $$ \lambda\in\Lambda:=\Big]\frac{\zeta^p \omega_\tau}{ \| \alpha \|_{B (0, \tau/2)} F(\zeta)} \big\{ \frac{\sigma(N,p)}{p \tau^p} + \frac{g(p,N)}{p} \big\},\, \frac{c^p }{p ( \frac{2p}{p-N} )^p \|\alpha\|_1 \max_{|t| \leq c} F(t)}\Big[, $$ we set \begin{equation}\label{e7} \delta_{\lambda,h}:=\min\big\{\frac{c^p - \lambda \sup_{|t|\leq c}F(t)}{H^c},\ \frac{1 - \lambda F(\zeta)} {p H_\zeta}\big\} \end{equation} and \begin{equation}\label{e8} \overline{\delta}_{\lambda, h}:=\min\big\{\delta_{\lambda, g},\ \frac{1}{\max\{0,\|\beta\|_1\} \limsup_{|t|\to\infty}\frac{H(t)}{t^p }\}}\big\}, \end{equation} where we define $r/0=+\infty$, so that, for instance, $\overline{\delta}_{\lambda, h}=+\infty$ when \[ \limsup_{|t|\to\infty}\frac{\|\beta\|_1 H(t)}{t^p}\leq 0, \] and $H_c=H^\zeta=0$. Now, we formulate our main result. \begin{theorem}\label{thm3} Assume that there exist constants $c > 0$ and $\zeta > 0$ with \[ c^p < \Big( \frac{2p}{p - N} \Big)^p \zeta^p \omega_\tau \Big[ \frac{\sigma(N,p)}{\tau^p} + g(p,N) \Big] \] such that \begin{itemize} \item[(A2)] $\frac{ \sup_{|t|\leq c} F(t)} {c^p}< \kappa F(\zeta)$; \item[(A3)] $\limsup_{|t|\to +\infty}\frac{\|\alpha\|_1 F(t)}{t^p}\leq0$. \end{itemize} Then, for each $$ \lambda\in\Lambda:=\Big]\frac{\zeta^p \omega_\tau}{ \| \alpha \|_{B (0, \tau/2)} F(\zeta)} \big\{ \frac{\sigma(N,p)}{p \tau^p} + \frac{g(p,N)}{p} \big\},\, \frac{c^p }{p \big( \frac{2p}{p-N} \big)^p \|\alpha\|_1 \max_{|t| \leq c} F(t)}\Big[ $$ and for every function $h: \mathbb{R}\to \mathbb{R}$ satisfying the condition $$ \limsup_{|t|\to\infty}\frac{\|\beta\|_1 H(t)}{t^p}<+\infty, $$ there exists $\overline{\delta}_{\lambda, h}>0$ given by \eqref{e8} such that, for each $\mu\in[0,\overline{\delta}_{\lambda, h}[$, problem \eqref{e1} admits at least three distinct weak solutions in $X$. \end{theorem} \begin{proof} To apply Theorem \ref{t1} to our problem, we introduce the functionals $\Phi, \Psi:X \to \mathbb{R} $ for each $u\in X$, as follows \begin{gather*} \Phi(u)=\frac{1}{p}\|u\|_r^p, \\ \Psi(u)=\int_{\mathbb{R}^N}\big[\alpha(x)F(u(x))+\frac{\mu}{\lambda}\beta(x)H(u(x)) \big]dx. \end{gather*} Now we show that the functionals $\Phi$ and $\Psi$ satisfy the required conditions. It is well known that $\Psi$ is a differentiable functional whose differential at the point $u\in X$ is $$ \Psi'(u)(v)=\int_{\mathbb{R}^N}[\alpha(x)f(u(x)) +\frac{\mu}{\lambda}\beta(x)h(u(x))]v(x)dx, $$ for every $v\in X$, as well as, is sequentially weakly upper semicontinuous. Furthermore, $\Psi':X \to X^{*}$ is a compact operator. Moreover, $\Phi$ is continuously differentiable and whose differential at the point $u\in X$ is \[ \Phi'(u)v =\int_{\mathbb{R}^N} |\nabla u(x)|^{p-2} \nabla u(x) \cdot \nabla v(x) dx + \int_{\mathbb{R}^N} |u(x)|^{p-2}u(x)v(x)dx, \] for every $v\in X$, while by standard arguments, one has that $\Phi$ is G\^{a}teaux differentiable and sequentially weakly lower semicontinuous, and its G\^{a}teaux derivative $\Phi'$ admits a continuous inverse on $X^{*}$. Clearly, the weak solutions of the problem \eqref{e1} are exactly the solutions of the equation $\Phi'(u)-\lambda\Psi'(u)=0$. Put $r=\frac{c^p}{\left( \frac{2p}{p-N} \right)^p p}$ and \begin{equation}\label{e9} w(x)= \begin{cases} 0, &x\in \mathbb{R}^N \setminus B(0,\tau),\\ \frac{2\zeta}{\tau} \left( \tau - |x| \right), &x\in B(0,\tau)\setminus B(0,\tau/2),\\ \zeta, &x \in B(0,\tau/2). \end{cases} \end{equation} It is easy to see that $w\in X$ and \[\|w\|_r^p = \zeta^p \omega_\tau \big[ \frac{\sigma(N,p)}{\tau^p} + g(p,N) \big]. \] Indeed \begin{align*} \int_{\mathbb{R}^N} & |\nabla w(x)|^p dx = \int_{B(0,\tau) \setminus B(0, \tau/2)} \frac{2^p \zeta^p}{\tau^p} dx\\ & = \frac{2^p \zeta^p}{\tau^p} (\operatorname{meas}(B(0,\tau))) - \operatorname{meas}(B(0,\tau/2)) = \frac{2^{p-N} \zeta^p \omega_\tau}{\tau^p} (2^N - 1), \end{align*} and \begin{gather*} \int_{\mathbb{R}^N} |w(x)|^p dx = \int_{B(0,\tau/2)} \zeta^p dx + \int_{B(0,\tau/2)} \frac{2^p \zeta^p}{\tau^p} (\tau - |x|)^p dx\\ \zeta^p \Big( \int_{B(0,\tau/2)} dx + \frac{2^p}{\tau^p} \int_{B(0,\tau) \setminus B(0,\tau/2)} (\tau - |x|)^p dx \Big) = \omega_\tau \zeta^p g(p,N). \end{gather*} Note that the last equality holds owing to \begin{equation}\label{eq:3.3} I_p := \int_{B(0,\tau)\setminus B(0,\tau/2)} (\tau - |x|)^p dx = N \omega_\tau \tau^p B_{(1/2,1)} (N, p+1). \end{equation} The easiest way to compute this integral is to go through a general polarcoordinates transformation. Let \begin{gather*} x_1 = \rho \cos\theta_1,\\ x_j = \rho \cos\theta_1 \cos\theta_2 \cdots \cos \theta_{j_1} \sin \theta_j, \quad (j = 2, \cdots, N-1) \\ x_N = \rho \cos\theta_1 \cos\theta_2 \cdots \cos \theta_{N_1}, \quad\text{for }\rho \in [\bar{\mu} \tau, \tau], \; \theta_j \in (-\pi/2, \pi/2],\\ j = 1, \cdots, N-2\text{ and }\theta_{N-1} \in (-\pi, \pi]. \end{gather*} The Jacobian of this transformation is \[ dx_1 \cdots dx_N = \rho^{N-1} \Big\{ \prod_{j=1}^{N-1} |\cos\theta_j|^{N-j-1} \Big\} d\rho d \theta_1 \cdots d\theta_{N-1}. \] Hence, one has \[ I_p = \Big( \int_{\tau/2}^{\tau} (\tau - \rho)^p \rho^{N-1} d \rho \Big) \Big( \int_{-\pi}^{\pi} d \theta_{N-1} \Big) \prod_{j=1}^{N-2} \int_{-\pi/2}^{\pi/2} |\cos\theta_j|^{N - j -1} d \theta_j. \] On the other hand, since \[ \prod_{j=1}^{N-2} \int_{-\pi/2}^{\pi/2} |\cos\theta_j|^{N - j -1} d \theta_j = \prod_{j = 1}^{N-2} \Gamma \big( \frac{N-j}{2} \big) \Gamma ( \frac{1}{2} )/ \Gamma \big( \frac{N-j+1}{2} \big), \] taking into account that \[ \prod_{j=1}^{N-2} \Gamma \big( \frac{N-j}{2} \big) \Gamma ( \frac{1}{2} )/ \Gamma \big( \frac{N-j+1}{2} \big) = \frac{N \pi^{N/2 -1}}{2 \Gamma ( \frac{N}{2} + 1 )}, \] an elementary computation gives \eqref{eq:3.3}. Moreover, from the condition $$ c^p < \zeta^p \omega_\tau \Big[ \frac{\sigma(N,p)}{\tau^p} + g(p,N) \Big] \Big( \frac{2p}{N-p} \Big)^p $$ one has $0\frac{1}{\lambda}. $$ Then \begin{equation}\label{e14} \frac{\int_{\mathbb{R}^N} \sup_{|t|\leq c}\alpha(x) F(t)dx+\frac{\mu}{\lambda}H^{c}} {\frac{c^p}{( \frac{2p}{N-p} )^pp} } <\frac{1}{\lambda}<\frac{\int_{\mathbb{R}^N}\alpha(x)F(w(x))dx +\|\beta\|_1 \frac{\mu}{\lambda} H_{\eta}} {\frac{\zeta^p \omega_\tau [ \frac{\sigma(N,p)}{\tau^p} + g(p,N) ]}{p}}. \end{equation} Hence from \eqref{e12}-\eqref{e14}, the condition (a1) of Theorem \ref{t1} is verified. Finally, since $\mu<\overline{\delta}_{\lambda, g}$, we can fix $l>0$ such that $$ \limsup_{|t|\to\infty}\frac{\|\beta\|_1 H(t)}{t^p}